A forest formula for pre-Lie exponentials, Magnus' operator and cumulant-cumulant relations

TL;DR

This paper introduces forest formulas for pre-Lie algebra computations, extending quantum field theory techniques to improve understanding of the exponential, Magnus operator, and cumulant relations in algebraic and probabilistic contexts.

Contribution

It generalizes forest formulas to pre-Lie algebras, providing new computational tools inspired by quantum field theory and applications in free probability.

Findings

Derived combinatorial formulas for cumulants in non-commutative probability.

Applied forest formulas to the pre-Lie exponential and Magnus operator.

Enhanced computational methods for pre-Lie algebra structures.

Abstract

Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same ideas and techniques generalize and provide effective tools to handle computations in these algebras, which are of utmost importance in numerical analysis and related areas. We illustrate our results by studying the action of the pre-Lie exponential and the Magnus operator in the free pre-Lie algebra and in a pre-Lie algebra of words originating in free probability. The latter example provides combinatorial formulas relating the different brands of cumulants in non-commutative probability.